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The mixed Hodge structure of an isolated critical point of a holomorphic function

  • V. I. Arnold
  • S. M. Gusein-Zade
  • A. N. Varchenko
Chapter
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

A mixed Hodge structure in a vector space is two filtrations of the space, satisfying the axioms indicated below. In the space of cohomologies, vanishing at the critical point of the holomorphic function, there is a natural mixed Hodge structure. The role of the above-mentioned filtrations is played by the weight and Hodge filtrations, introduced in Chapter 13. The weight filtration is constructed from the Jordan structure of the monodromy operator and reflects the behaviour of integrals over vanishing cycles under analytic continuation of the integrals round critical values of the parameter.

Keywords

Holomorphic Function Principal Part Hodge Structure Bernstein Polynomial Holomorphic Form 
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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • V. I. Arnold
  • S. M. Gusein-Zade
    • 1
  • A. N. Varchenko
    • 2
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Department of MathematicsThe University of North Carolina at Chapel HillChapel HillUSA

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